Ring Function

  • Central Forces 2021 Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.
    1. Find \(N\).

    2. Plot this wave function.
    3. Plot the probability density.
    4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
    5. What is the expectation value of \(L_z\) in this state?